The Self Purification Model for Water Pollution

International Journal Of Mathematics And Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
Www.Ijmsi.org || Volume 3 Issue 1 || January. 2015 || PP-17-32
www.ijmsi.org 17 | P a g e
The Self Purification Model for Water Pollution
Khemlal Mahto1
, Indeewar Kumar2
1
(Department of Mathematics,University College of Engineering &Technology (UCET), VBU, Hazaribah,
Jharkhand, India)
2
(Department of Mathematics, Manipal University Jaipur, Rajasthan, India)
ABSTRACT : Mathematical model have been developed for self purification of river water. Since the
ecology of the river depends largely on the quantity of dissolved oxygen in its water, this dissolved oxygen (DO)
seems to be a convenient criterion for measuring the degree of pollution of a river as far as organic pollution is
concerned. However, even the term organic pollution embodies a great number of different materials and the
question assessing this load of pollutants is raised.Considering that the effect of all kinds of organic matter will
be consumption of dissolved oxygen, it is usual to measure the load of organic pollution by the quantity of
oxygen necessary to completely oxidize this load by bacteriological breakdown, i.e. by its biological oxygen
demand (BOD).The present analysis deals with the polluted water in a river. Mathematical models is formulated
which simulates the variations of the parameters D = DO (Dissolved oxygen and B = BOD (Biological oxygen
demand) over time at each point of a river (or reach of a river).The validity of the prospection use of the model
depends heavily on the validity of the equations which have been used and this depends on knowledge of
accurate hydrolic parameters advection, diffusion and reaeration. These parameters are fairly well known by a
theoretical approach when compared to biodegradation and other phenomena. A details comprehensive field
measurement survey is necessary to determine empirically the bulk biodegradation coefficients to be introduce
into the model. Under these conditions a complete understanding of the mechanism of self purification can be
obtained. This mathematical models is very helpful for the study of oxygen in rivers.
KEYWORDS: Dissolved Oxygen, Molecular Diffusion, Surface Reaeration, Water Quality Modeling
I. INTRODUCTION
Choosing a criterion to characterize the degree of pollution of river is not easy because several quite
different kinds of pollution (organic matter, radioactivity, heat, poisonous chemicals, pathogenic germs) are
found in a river. The effect also are quit different, for example turbidity, death of the fauna, or eutrophication. In
fact, pollution of rivers by organic matter (most municipal and industrial effluents) is certainly the most
significant in terms of quantity and by effect on the river. In river water organic matter is naturally eliminated by
a biochemical degradation performed by bacteria. As long as dissolved oxygen is available, the biochemical
breakdown can be considered equivalent to an oxidization reaction which will lower the level of dissolved
oxygen and therefore deteriorate the ecological balance in the river. When there is no oxygen left in the water,
the breakdown of organic matter becomes anaerobic. Fortunately, atmospheric oxygen enters the water through
the surface, and thus a river is capable of eliminating a definite amount of organic matter by itself, this ability of
the river to regenerate is called „Self purification‟.
Since the ecology of the river depends largely on the quantity of dissolved oxygen in its water, this
dissolved oxygen (DO) seems to be a convenient criterion for measuring the degree of pollution of a river as far
as organic pollution is concerned. However, even the term organic pollution embodies a great number of
different materials and the question assessing this load of pollutants is raised.Considering that the effect of all
kinds of organic matter will be consumption of dissolved oxygen, it is usual to measure the load of organic
pollution by the quantity of oxygen necessary to completely oxidize this load by bacteriological breakdown, i.e.
by its biological oxygen demand (BOD)Chevereau formulated Mathematical Models for oxygen balance in
rivers. Streeter and Phelps obtained pollution equation by making important assumptions for the investigation of
biochemical breakdown camp. Elder has proposed value of the semiempirical coefficients. A for a very wide
open channel. He has taken A = 0.067, 0.23 and 10 for the mean vertical diffusion coefficient over depth, mean
transverse diffusion coefficient and for the mean longitudinal diffusion coefficient respectively. O Conner and
Dobbins developed a formula for surface reaeration coefficient based on a theoretical analysis which yields
good result. Gannon founds k1 coefficients ranging from 0.01 to 1.0 day -1 . He also advocated two step explicit
method.

The Self Purification Model for Water…
In natural stream, molecular diffusion is very weak compared with turbulent diffusion and therefore
only turbulent diffusion is considered. The study of diffusion goes back to Prandtl‟s first experiment turbulence
studies. The basis of prandtl‟s turbulence theory are clearly shown in Bakhmelff‟s book. The turbulent diffusion
in a very wide rectangle channel assuming two dimensional flow was investigated by Elder.The present analysis
deals with the polluted water in a river. Mathematical models is formulated which simulates the variations of the
parameters D = DO (Dissolved oxygen and B = BOD (Biological oxygen demand) over time at each point of a
river (or reach of a river).
Symbols
A = An empirical coefficient
B = (BOD) Biological oxygen demand.
COD = Chemical oxygen demand.
C = Concentration of dissolved oxygen of the particle.
CA = The rate of input/output of DO
CS = The Saturation Concentration of oxygen
D = Oxygen deficit
DO = The value of the D at time = o
DC = Critical value of D
D1 = The longitudinal diffusion coefficient
DO = Dissolved oxygen
g = The weight acceleration
h = The Height of the fall = the depth
k = A coefficient depending on the shape of weir and on quality of the water.
K1 , K2 = Biological coefficients
L = Polluted water in a river with a concentration of organic pollution
LA = The rate of input / output of BOD
LO = Value of time t = O
S = Slope of the energy line
t = Time
T = The water temperature
U = The mean velocity
u* = the wall velocity
UDO = Ultimate oxygen demand
X = The abscissa along the river.
II. EQUATIONS AND MODELING
Pollution Equations:Consider a particle of polluted water in a river with a concentration of organic pollution
B(=BOD) denoted by L. This particle undergoes several processes which will change its concentration L.
Among these, the main ones are advection, diffusion, biodegradation of organic matter, and other inputs of
organic matter. Let C be the concentration of dissolved oxygen of particle. The processes which will change C
are mainly advection, diffusion oxidization of organic matter, reaeration through the surface, and other inputs or
outputs of organic matter.
A mathematical model for organic pollution concentration L describing the variations over time is
(1)
And the equation describing the oxygen concentration C is
(2)
Equation (1) describes organic pollution where (i) C and L are assumed to be uniformed in the cross
section and (ii) is constant over time and the flow assumed steady. In the Streeter and Phelps equation the
biochemical breakdown term is written

This shows that the pollution load measured by its total BOD (BOD) is degraded according to an
exponential law (Fig-1) at a constant rate, K1 called biodegraded coefficient
Replacing e by 10,
The equation (5) becomes
L = LO 10-K1t (6)
provided that K1 = 2.3 KO , Streeter and Phelps estimated, based on laboratory experiments that KO
was approximately equal to 0.1 day -1 . Under these conditions, there is a relation between the initial pollution
load LO measured by B = [BOD] and the load Lt after a time t of incubation; for instance, if t = 5 days,
Then
BO D = 1.45 B O D 5
Therefore, the determination of B = BO D which theoretically needs an infinite time (which is
impossible), can be obtained by determining the BOD ( B5 ) by way of laboratory BOD ( =B5 ) tested and
applying equation (7)
Surface Reaeration: In the Streeter Phelps equation, this term is written as
K2 (CS – C) (8)
Where K2 is a coefficient which depends on local hydraulic conditions.
The molecular diffusion of dissolved oxygen in water is very small (2.10-5 m2/s). But it is generally
acknowledged that except close to the surface and to the river bed, turbulent diffusion is sufficient to mix the
dissolved oxygen in uniform manner over a vertical section. Field measurement support this statement, so
K 2 = (9)
In which m would be approximately 1 and n approximately is 1.5. Most author consider C as a
Constant coefficient, a coefficient depending on the diffusion coefficient, or on Froulde‟s number. With m = 1, n
= 1.5 and C being a constant coefficient a “mean formula” is obtained which is satisfactory in practical cases. It
is generally admitted that the coefficient K2 is dependent on water temperature according to the following
equation.
K 2 (T) = K 2 (200° C) ө (T-20)
In which ө is approximately 1.025. But this value of ө is not the most accurate value; but pollution
models do not need such accuracy.
For unsteady flow, equation (1) can be written as
(10)
Where S is slope of energy line.
Or (11)
If U and S satisfy the mass equation then

Using (12) in (11), we get
(13)
Analytical Solution: Analytical solution can help in understanding the mechanism of self purification in river
and the influence of the main parameters. Consider the case in which the variations concentration in BOD (=B)
and oxygen are due only to biodegration and reaeration through the surface. Assume that the flow is steady and
that diffusion is ignored. If K1 and K2 are constants, the concentration in oxygen C of a Particle moving at
velocity U is given by
`
(14)
Assume, D = CS – C,
Where D is the oxygen deficit
So (14) becomes
Or
Or (15)
I.F.= e = e K 2t
The solution of (15) is given by
(16)
inserting L from (5)
(17)
Where E is constant of integration initially, t = 0, D = D0 . Therefore,
from (17), we have
Putting E in (17), we get

(18)
Here L0 ; D0 ; denote the value of L and D at time t = 0
Using k1 and k2 in place of K1, K2 and replacing e by 10, we get
(19)
The graphical representation of this equation is shown as a sag curve in Fig 2. Worst condition in a
river are observed at a point where the dissolved oxygen concentration is a minimum, therefore the point of the
centre when D is equal to a maximum DC and called the critical deficit, appears at the critical time tC ,so for
critical deficit DC and critical time tC.
Now, differentiation (19) w.r. to t, we get
(20)
(21)
Again differentiating, we have
Or

(22)
When , from equation (21), we have
Or
Or
Or (23)
Or
Or
Or (24)
Or
Or (25)
Taking log both sides,
= (k2 – k1 ) t. log10 10
Or
(26)
ThereforeTaking k1 = 0.1 in (26)
(27)

Special Cases : For different values of , the values of tc from (27) are as follows:
(28)
(29)
(30)
(31)
(32)
(33)
Now, from the equation (19)

(34)
Writing Dc for D and tc for t, we have
(35)
Or (36)
Taking k1 = 0.1
(37)
Now, equation (22) is
(use of 25)

Or (38)
(Use of 26)

Hence the value of D is maximum for the value of t given in (26). Thus Dc is obtained with the aid of t.
III. RESULTS AND CONCLUSION
Expression (19) for oxygen deficit D has been obtained in terms of biological coefficients k1 and k2.
The expressions for critical Dc (35) and critical time tc (27) are also derived.
Streeter and Phelps estimated that k1 = 0.1 day -1 . On this basis calculations for L/L0 and Dc / L0 have
been made. Result of calculation are entered in tables (1) and (2) and graphically shown by curve in Fig 3 and
(4). Also k2tc has been numerically computed for different values when k1 = 0.1. Results are entered in tables
(3) and illustrated graphically in figure (5)
From analysis of the numerical computations we observe that
a. Dc / L0 increases as Dc / L0 increases (see Fig 4)
b. As Dc / L0 increases k2tc decreases. (see Fig 5)
The validity of the prospection use of the model depends heavily on the validity of the equations
which have been used and this depends on knowledge of accurate hydrolic parameters advection, diffusion and
reaeration. These parameters are fairly well known by a theoretical approach when compared to biodegradation
and other phenomena. A details comprehensive field measurement survey is necessary to determine empirically
the bulk biodegradation coefficients to be introduce into the model. Under these conditions a complete
understanding of the mechanism of self purification can be obtained. This mathematical models is very helpful
for the study of oxygen in rivers.
Table-1
Mathematical Model for Self Purification in River
Variance of L/L0 against t.
t = 10-0.23t
1 0.5588
2 0.3467
3 0.2041
4 0.1202
5 0.0707
6 0.04168
7 0.0245
<0

Table-2
Mathematical Model for Self Purification in River
Variation of Dc / L0 against k1/k2 for different value of Dc / L0, using equation (27) & (36)
Dc/L0
D0 /L0
k1/k2
0.0 0.5 1.0 2.0 5.0
0.01 0.0098
0.04 0.0377
0.08 0.0727
0.1 0.0894
0.2 0.1679 0.149
0.4 0.3067 0.3491
0.6 0.4301 0.5602 0.061
0.8 0.5429 0.6846 0.894
1.0 1 1 1 1
2.0 1.0953 1.3296 1.5577 3.2515
4.0 1.7923 2.1553 2.4783 3.6211
6.0 2.358 2.8304 3.2357 5.5551
8.0 2.85 3.4125 3.9326 6.567
10.0 3.2919 3.9384 4.487 7.494
Table – 3
Mathematical model for self purification in river
Value of k2tc for different value of D0 / L0, using equation (27)
D0 /L0
k1/k2
0.0 0.5 1.0 2.0 5.0
0.01 2.0202
0.02 1.7336
0.04 1.4561
0.08 1.1922
0.1 1.1111
0.2 0.8737 1.4701
0.4 0.6632 0.3401
0.6 5546 0.1845
0.8 0.4845 0.1143 0.1401 -1.0205
2.0 0.301 0.2041 0.1249 0 -0.243
4.0 0.2006 0.1545 0.068 0.0248
6.0 0.1561 0.1252 0.1029 0.0704 0.013
8.0 0.129 0.1064 0.0887 0.0662 0.0246
10.0 0.1111 0.0931 0.0801 0.0614 0.0288

Fig. 1Biodegration against t
Figure 2 Dissolved Oxygen Sag-Curve
--------------------------------------------------------
D

Figure 3 Graph of L/Lo against t
Relation Between Dc
/ L0
& k1
/ k2
Graph of
L / Lo
against t

Figure 4 Relation Between Dc/Lo & K1/K2
Dc
/
L0

Figure 5 Graph of K2tc against k1/k2
ACKNOWLEDGEMENTS
I am very much thankful to professor Dr. B.N. Mishra, Rred. University prof. and Head, Deptt. of Mathematics,
VinobaBhave University, Hazaribag for suggestions and improvement.
REFERENCES
[1] Dobbins (R.A. (ed) (1979): Atmospheric motion and air pollution ,Jhon Wiley and Sons. New York
[2] Dobbins, W.E.(1964):BOD and Oxygen Relationship in streams, “J. San. Eng. Div. Proc Amer.Soc Civil Engr. 90(SA3) 53 (1964)
[3] Dysart, B.C. and Heins, W.W. (1970): Control of water quality in a complex natural system "IEEE Transactions on system Science
and Cybernetic.
[4] Dobbins, W.E. and Bella, D.A. (1968): Difference Modeling of stream pollution, “ J. San Eng. DivProc ASCE 94 SAS:
[5] Klein, L.(1965):River pollution 2. Causes and Effect Butterworth, London XIV + 456 pp.
[6] Morly D.A. (1979):Mathematical modeling in water and waste water treatment. Intel Ideas Philadelphia. P.A.
[7] O' Conner, J (1967):“The Temporal and Special Distribution of Dissolve Oxygen in streams, “water Res. Res 3(1): 65
[8] Streeter, H.W. & E.B. Phelps (1925):“A Study of Pollution and natural purification of water of the Ohio river” U.S. Public Health
Bulletin 146
Figure- 5
Graph of k2
tc
against k1
/k2
---------k1
/ k2
----------

The Self Purification Model for Water Pollution

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Similaire à The Self Purification Model for Water Pollution

Similaire à The Self Purification Model for Water Pollution (20)

Dernier

Dernier (20)

The Self Purification Model for Water Pollution